# How can l prove that Newton's laws are time invariant?

• stefano77
In summary, Newton's law can be proven to be time invariant by showing that if x(t) is a solution of dd/ddx x(t) = f(x(t)), then y(t) = x(-t) is also a solution of dd/ddt y(t) = f(y(t)). This can be further demonstrated by noting that the second derivative with respect to time for y(t) is equal to the second derivative with respect to time of x(-t), which is equivalent to f(x(-t)).

#### stefano77

Misplaced Homework Thread -- Moved to the Schoolwork forums by the Mentors
how can l prove Newton's law is time invariant?

if x (t) is a solution of dd/ddx x(t) = f(x(t)) then if l put y(-t) dd/ddt y(t)=dd/ddt x(-t). Now how dd/ddt x(-t) is equal to f(x(-t))?

dd/ddt is second derivative with respect to time

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• topsquark and vanhees71
stefano77 said:
how can l prove Newton's law is time invariant?

if x (t) is a solution of dd/ddx x(t) = f(x(t)) then if l put y(-t) such that dd/ddt y(t)=dd/ddt x(-t). Now how dd/ddt x(-t) is equal to f(x(-t))?

dd/ddt is second derivative with respect to time
##\dfrac{d^2}{dt^2} y(-t) = \dfrac{d^2}{dt^2} x(t)##

or equivalently
##\dfrac{d^2}{d(-t)^2} y(t) = \dfrac{d^2}{d(-t)^2} x(-t)##

Can you finish?

-Dan